Question: Solve for $x$ : $ 8|x + 7| + 5 = 5|x + 7| + 7 $
Subtract $ {5|x + 7|} $ from both sides: $ \begin{eqnarray} 8|x + 7| + 5 &=& 5|x + 7| + 7 \\ \\ { - 5|x + 7|} && { - 5|x + 7|} \\ \\ 3|x + 7| + 5 &=& 7 \end{eqnarray} $ Subtract ${5}$ from both sides: $ \begin{eqnarray} 3|x + 7| + 5 &=& 7 \\ \\ { - 5} &=& { - 5} \\ \\ 3|x + 7| &=& 2 \end{eqnarray} $ Divide both sides by ${3}$ $ \dfrac{3|x + 7|} {{3}} = \dfrac{2} {{3}} $ Simplify: $ |x + 7| = \dfrac{2}{3}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 7 = -\dfrac{2}{3} $ or $ x + 7 = \dfrac{2}{3} $ Solve for the solution where $x + 7$ is negative: $ x + 7 = -\dfrac{2}{3} $ Subtract ${7}$ from both sides: $ \begin{eqnarray} x + 7 &=& -\dfrac{2}{3} \\ \\ {- 7} && {- 7} \\ \\ x &=& -\dfrac{2}{3} - 7 \end{eqnarray} $ Change the ${ - 7}$ to an equivalent fraction with a denominator of $3$ $ x = - \dfrac{2}{3} {- \dfrac{21}{3}} $ $ x = -\dfrac{23}{3} $ Then calculate the solution where $x + 7$ is positive: $ x + 7 = \dfrac{2}{3} $ Subtract ${7}$ from both sides: $ \begin{eqnarray} x + 7 &=& \dfrac{2}{3} \\ \\ {- 7} && {- 7} \\ \\ x &=& \dfrac{2}{3} - 7 \end{eqnarray} $ Change the ${ - 7}$ to an equivalent fraction with a denominator of $3$ $ x = \dfrac{2}{3} {- \dfrac{21}{3}} $ $ x = -\dfrac{19}{3} $ Thus, the correct answer is $x = -\dfrac{23}{3} $ or $x = -\dfrac{19}{3} $.